![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mvrcl | Structured version Visualization version GIF version |
Description: A power series variable is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.) |
Ref | Expression |
---|---|
mvrcl.s | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mvrcl.v | ⊢ 𝑉 = (𝐼 mVar 𝑅) |
mvrcl.b | ⊢ 𝐵 = (Base‘𝑃) |
mvrcl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
mvrcl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mvrcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
Ref | Expression |
---|---|
mvrcl | ⊢ (𝜑 → (𝑉‘𝑋) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2728 | . . 3 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
2 | mvrcl.v | . . 3 ⊢ 𝑉 = (𝐼 mVar 𝑅) | |
3 | eqid 2728 | . . 3 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
4 | mvrcl.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | mvrcl.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
6 | mvrcl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
7 | 1, 2, 3, 4, 5, 6 | mvrcl2 21928 | . 2 ⊢ (𝜑 → (𝑉‘𝑋) ∈ (Base‘(𝐼 mPwSer 𝑅))) |
8 | fvexd 6912 | . . 3 ⊢ (𝜑 → (𝑉‘𝑋) ∈ V) | |
9 | eqid 2728 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
10 | eqid 2728 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
11 | 1, 9, 10, 3, 7 | psrelbas 21878 | . . . 4 ⊢ (𝜑 → (𝑉‘𝑋):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
12 | 11 | ffund 6726 | . . 3 ⊢ (𝜑 → Fun (𝑉‘𝑋)) |
13 | fvexd 6912 | . . 3 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
14 | snfi 9068 | . . . 4 ⊢ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ∈ Fin | |
15 | 14 | a1i 11 | . . 3 ⊢ (𝜑 → {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ∈ Fin) |
16 | eqid 2728 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
17 | eqid 2728 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
18 | 4 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝐼 ∈ 𝑊) |
19 | 5 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑅 ∈ Ring) |
20 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑋 ∈ 𝐼) |
21 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) | |
22 | eldifsn 4791 | . . . . . . . 8 ⊢ (𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) ↔ (𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑥 ≠ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) | |
23 | 21, 22 | sylib 217 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑥 ≠ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) |
24 | 23 | simpld 494 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) |
25 | 2, 10, 16, 17, 18, 19, 20, 24 | mvrval2 21924 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → ((𝑉‘𝑋)‘𝑥) = if(𝑥 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), (1r‘𝑅), (0g‘𝑅))) |
26 | 23 | simprd 495 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑥 ≠ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) |
27 | 26 | neneqd 2942 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → ¬ 𝑥 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) |
28 | 27 | iffalsed 4540 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → if(𝑥 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅)) |
29 | 25, 28 | eqtrd 2768 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → ((𝑉‘𝑋)‘𝑥) = (0g‘𝑅)) |
30 | 11, 29 | suppss 8198 | . . 3 ⊢ (𝜑 → ((𝑉‘𝑋) supp (0g‘𝑅)) ⊆ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) |
31 | suppssfifsupp 9403 | . . 3 ⊢ ((((𝑉‘𝑋) ∈ V ∧ Fun (𝑉‘𝑋) ∧ (0g‘𝑅) ∈ V) ∧ ({(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ∈ Fin ∧ ((𝑉‘𝑋) supp (0g‘𝑅)) ⊆ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑉‘𝑋) finSupp (0g‘𝑅)) | |
32 | 8, 12, 13, 15, 30, 31 | syl32anc 1376 | . 2 ⊢ (𝜑 → (𝑉‘𝑋) finSupp (0g‘𝑅)) |
33 | mvrcl.s | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
34 | mvrcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
35 | 33, 1, 3, 16, 34 | mplelbas 21932 | . 2 ⊢ ((𝑉‘𝑋) ∈ 𝐵 ↔ ((𝑉‘𝑋) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (𝑉‘𝑋) finSupp (0g‘𝑅))) |
36 | 7, 32, 35 | sylanbrc 582 | 1 ⊢ (𝜑 → (𝑉‘𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 {crab 3429 Vcvv 3471 ∖ cdif 3944 ⊆ wss 3947 ifcif 4529 {csn 4629 class class class wbr 5148 ↦ cmpt 5231 ◡ccnv 5677 “ cima 5681 Fun wfun 6542 ‘cfv 6548 (class class class)co 7420 supp csupp 8165 ↑m cmap 8844 Fincfn 8963 finSupp cfsupp 9385 0cc0 11138 1c1 11139 ℕcn 12242 ℕ0cn0 12502 Basecbs 17179 0gc0g 17420 1rcur 20120 Ringcrg 20172 mPwSer cmps 21836 mVar cmvr 21837 mPoly cmpl 21838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-tset 17251 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-mgp 20074 df-ur 20121 df-ring 20174 df-psr 21841 df-mvr 21842 df-mpl 21843 |
This theorem is referenced by: mvrf2 21934 subrgmvrf 21971 mplcoe3 21975 mplcoe5lem 21976 mplcoe5 21977 mplcoe2 21978 mplbas2 21979 evlsvarpw 22039 mpfproj 22047 mpfind 22052 mhpvarcl 22071 vr1cl 22135 evlsvarval 41798 selvcllem5 41815 selvvvval 41818 |
Copyright terms: Public domain | W3C validator |